What is the set of homotopy classes of maps of $X$ into $Y$?

Question

Munkres's Topology denotes the set of homotopy classes of maps of $X$ into $Y$ by $[X,Y]$. How do we write $[X,Y]$ in the sense of set, i.e., $[X,Y]=\{[f] \in ... | ... \}$? Is an element of $[X,Y]$ a class of continuous maps from $X$ to $Y$?

Answer 1

You start with the set of all continuous functions:

$$C(X,Y)=\{f:X\to Y\ |\ f\text{ is continuous}\}$$

then you define a relation:

$$f\sim g\text{ if and only if }f\text{ is homotopic to }g$$

and finally you define the quotient set

$$[X,Y]=C(X,Y)/\sim$$

Elements of $[X,Y]$ are sets which are equivalence classes

$$[f]_{\sim}=\{g\in C(X,Y)\ |\ g\sim f\}$$ $$[X,Y]=\{[f]_{\sim}\ |\ f\in C(X,Y)\}$$

For example if $Y=\mathbb{R}$ (or any other contractible space) then $[X,Y]$ has only one point.

Another example is $[S^1, S^1]$ which is equinumerous to $\mathbb{Z}$.